Editing Block matrix (section)

===Multiplication===
It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "[[Conformable matrix|conformable]] partitions"<ref>{{cite book |last=Eves |first=Howard |author-link=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=[https://archive.org/details/elementarymatrix0000eves_r2m2/page/37 37] |url=https://archive.org/details/elementarymatrix0000eves_r2m2 |url-access=registration |edition=reprint |access-date=24 April 2013 |quote=A partitioning as in Theorem 1.9.4 is called a ''conformable partition'' of ''A'' and ''B''.}}</ref> between two matrices <math>A</math> and <math>B</math> such that all submatrix products that will be used are defined.<ref>{{cite book |last=Anton |first=Howard |title=Elementary Linear Algebra |year=1994 |publisher=John Wiley |location=New York |isbn=0-471-58742-7 |page=36 |edition=7th |quote=...provided the sizes of the submatrices of A and B are such that the indicated operations can be performed.}}</ref>
{{Cquote
| quote = Two matrices <math>A</math> and <math>B</math> are said to be partitioned conformally for the product <math>AB</math>, when <math>A</math> and <math>B</math> are partitioned into submatrices and if the multiplication <math>AB</math> is carried out treating the submatrices as if they are scalars, but keeping the order, and when all products and sums of submatrices involved are defined.
| author = Arak M. Mathai and Hans J. Haubold
| source = ''Linear Algebra: A Course for Physicists and Engineers''<ref>{{Cite book |last1=Mathai |first1=Arakaparampil M. |title=Linear Algebra: a course for physicists and engineers |last2=Haubold |first2=Hans J. |date=2017 |publisher=De Gruyter |isbn=978-3-11-056259-0 |series=De Gruyter textbook |location=Berlin Boston |pages=162}}</ref>
}}

Let <math>A</math> be the matrix defined in {{section link||Transpose}}, and let <math>B</math> be the matrix defined in {{section link||Addition}}. Then the matrix product

:<math>
  C = AB
</math>

can be performed blockwise, yielding <math>C</math> as an <math>(p \times s)</math> matrix. The matrices in the resulting matrix <math>C</math> are calculated by multiplying:

:<math>
  C_{ij} = \sum_{k=1}^{q} A_{ik}B_{kj}. 
</math><ref name=":3">{{Cite book |last=Johnston |first=Nathaniel |title=Introduction to linear and matrix algebra |date=2021 |publisher=Springer Nature |isbn=978-3-030-52811-9 |location=Cham, Switzerland |pages=30,425}}</ref>

Or, using the [[Einstein notation]] that implicitly sums over repeated indices:

:<math>
  C_{ij} = A_{ik}B_{kj}. 
</math>

Depicting <math>C</math> as a matrix, we have

:<math>C = AB = \begin{bmatrix}
\sum_{i=1}^{q} A_{1i}B_{i1} & \sum_{i=1}^{q} A_{1i}B_{i2} & \cdots & \sum_{i=1}^{q} A_{1i}B_{is} \\
\sum_{i=1}^{q} A_{2i}B_{i1} & \sum_{i=1}^{q} A_{2i}B_{i2} & \cdots & \sum_{i=1}^{q} A_{2i}B_{is} \\
\vdots & \vdots & \ddots & \vdots \\
\sum_{i=1}^{q} A_{pi}B_{i1} & \sum_{i=1}^{q} A_{pi}B_{i2} & \cdots & \sum_{i=1}^{q} A_{pi}B_{is}
\end{bmatrix}</math>.<ref name=":2" />